Transformation matrix
Encyclopedia
In linear algebra
, linear transformation
s can be represented by matrices. If T is a linear transformation mapping Rn to Rm and x is a column vector with n entries, then
for some m×n matrix A, called the transformation matrix of T.
There is an alternative expression of transformation matrices involving row vectors that is preferred by some authors.
Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on a n-dimensional Euclidean space
Rn, can be represented as linear transformations on the n+1-dimensional space Rn+1. These include both affine transformations (such as translation
) and projective transformations. For this reason, 4x4 transformation matrices are widely used in 3D computer graphics
. These n+1-dimensional transformation matrices are called, depending on their application, affine transformation matrices, projective transformation matrices, or more generally non-linear transformation matrices. With respect to a n-dimensional matrix, a n+1-dimensional matrix can be described as an augmented matrix
.
in functional form, it is easy to determine the transformation matrix A by simply transforming each of the vectors of the standard basis
by T and then inserting the results into the columns of a matrix. In other words,
For example, the function
is a linear transformation. Applying the above process (suppose that n = 2 in this case) reveals that
and 
. Written in matrix form, this becomes:
Similarly, for a rotation clockwise about the origin, the functional form is
and 
and the matrix form is:
(that is, enlarging or shrinking), we have
and 
. The matrix form is:
When
, then the matrix is a squeeze mapping
and preserves area
s in the plane.
and 
; the shear matrix
, applied to column vectors, is:
A shear parallel to the y axis has
and 
, which has matrix form:
be a vector in the direction of the line:
To reflect a point through a plane
(which goes through the origin), one can use 
, where 
is the 3x3 identity matrix and 
is the three-dimensional unit vector for the surface normal of the plane. If the L2 norm of 
and 
is unity, the transformation matrix can be expressed as:
Note that these are particular cases of a Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation; it is an affine transformation
.
be a vector in the direction of the line. Then use the transformation matrix:
As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation.
Parallel projection
s are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates must be used.
Composition is accomplished by matrix multiplication
. If A and B are the matrices of two linear transformations, then the effect of applying first A and then B to a vector x is given by:
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. Note that the multiplication is done in the opposite order from the English sentence: the matrix of "A followed by B" is BA, not AB.
A consequence of the ability to compose transformations by multiplying their matrices is that transformations can also be inverted by simply inverting their matrices. So, A-1 represents the transformation that "undoes" A.
s with matrices, we can use homogeneous coordinates
. This means representing a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher dimensions. Using this system, translation can be expressed with matrix multiplication. The functional form
; 
becomes:
All ordinary linear transformations are included in the set of affine transformations, and can be described as a simplified form of affine transformations. Hence, any linear transformation can be also represented by a general transformation matrix. The latter is obtained by expanding the corresponding linear transformation matrix by one row and column, filling the extra space with zeros except for the lower-right corner, which must be set to 1. For example, the clockwise rotation matrix from above becomes:
Using transformation matrices containing homogeneous coordinates, translations can be seamlessly intermixed with all other types of transformations. The reason is that the real plane is mapped to the w = 1 plane in real projective space, and so translation in real Euclidean space
can be represented as a shear in real projective space. Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates, it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a shear).
When using affine transformations, the homogeneous component of a coordinate vector (normally called w) will never be altered. One can therefore safely assume that it is always 1 and ignore it. However, this is not true when using perspective projections.
, is the perspective projection. Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection. This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer.
The simplest perspective projection uses the origin as the center of projection, and z = 1 as the image plane. The functional form of this transformation is then
; 
. We can express this in homogeneous coordinates
as:
After carrying out the matrix multiplication, the homogeneous component wc will, in general, not be equal to 1. Therefore, to map back into the real plane we must perform the homogeneous divide or perspective divide by dividing each component by wc:
More complicated perspective projections can be composed by combining this one with rotations, scales, translations, and shears to move the image plane and center of projection wherever they are desired.
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
s can be represented by matrices. If T is a linear transformation mapping Rn to Rm and x is a column vector with n entries, then
for some m×n matrix A, called the transformation matrix of T.
There is an alternative expression of transformation matrices involving row vectors that is preferred by some authors.
Uses
Matrices allow arbitrary linear transformations to be represented in a consistent format, suitable for computation. This also allows transformations to be concatenated easily (by multiplying their matrices).Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on a n-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn, can be represented as linear transformations on the n+1-dimensional space Rn+1. These include both affine transformations (such as translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
) and projective transformations. For this reason, 4x4 transformation matrices are widely used in 3D computer graphics
3D computer graphics
3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
. These n+1-dimensional transformation matrices are called, depending on their application, affine transformation matrices, projective transformation matrices, or more generally non-linear transformation matrices. With respect to a n-dimensional matrix, a n+1-dimensional matrix can be described as an augmented matrix
Augmented matrix
In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.Given the matrices A and B, where:A =...
.
Finding the matrix of a transformation
If one has a linear transformation in functional form, it is easy to determine the transformation matrix A by simply transforming each of the vectors of the standard basisStandard basis
In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...
by T and then inserting the results into the columns of a matrix. In other words,
For example, the function
is a linear transformation. Applying the above process (suppose that n = 2 in this case) reveals thatExamples in 2D graphics
Most common geometric transformations that keep the origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear. In two dimensions, linear transformations can be represented using a 2×2 transformation matrix.Rotation
For rotation by an angle θ counter clockwise about the origin, the functional form is and . Written in matrix form, this becomes:Similarly, for a rotation clockwise about the origin, the functional form is
and and the matrix form is:Scaling
For scalingScaling (geometry)
In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
(that is, enlarging or shrinking), we have
and . The matrix form is:When
, then the matrix is a squeeze mappingSqueeze mapping
In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a Euclidean motion.For a fixed positive real number r, the mapping →...
and preserves area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
s in the plane.
Shearing
For shear mapping (visually similar to slanting), there are two possibilities. For a shear parallel to the x axis has and ; the shear matrixShear matrix
In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another...
, applied to column vectors, is:
A shear parallel to the y axis has
and , which has matrix form:Reflection
To reflect a vector about a line that goes through the origin, let be a vector in the direction of the line:To reflect a point through a plane
(which goes through the origin), one can use , where is the 3x3 identity matrix and is the three-dimensional unit vector for the surface normal of the plane. If the L2 norm of and is unity, the transformation matrix can be expressed as:Note that these are particular cases of a Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation; it is an affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
.
Orthogonal projection
To project a vector orthogonally onto a line that goes through the origin, let be a vector in the direction of the line. Then use the transformation matrix:As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation.
Parallel projection
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
s are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates must be used.
Composing and inverting transformations
One of the main motivations for using matrices to represent linear transformations is that transformations can then be easily composed (combined) and inverted.Composition is accomplished by matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
. If A and B are the matrices of two linear transformations, then the effect of applying first A and then B to a vector x is given by:
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. Note that the multiplication is done in the opposite order from the English sentence: the matrix of "A followed by B" is BA, not AB.
A consequence of the ability to compose transformations by multiplying their matrices is that transformations can also be inverted by simply inverting their matrices. So, A-1 represents the transformation that "undoes" A.
Affine transformations
To represent affine transformationAffine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s with matrices, we can use homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
. This means representing a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher dimensions. Using this system, translation can be expressed with matrix multiplication. The functional form
; becomes:All ordinary linear transformations are included in the set of affine transformations, and can be described as a simplified form of affine transformations. Hence, any linear transformation can be also represented by a general transformation matrix. The latter is obtained by expanding the corresponding linear transformation matrix by one row and column, filling the extra space with zeros except for the lower-right corner, which must be set to 1. For example, the clockwise rotation matrix from above becomes:
Using transformation matrices containing homogeneous coordinates, translations can be seamlessly intermixed with all other types of transformations. The reason is that the real plane is mapped to the w = 1 plane in real projective space, and so translation in real Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
can be represented as a shear in real projective space. Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates, it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a shear).
When using affine transformations, the homogeneous component of a coordinate vector (normally called w) will never be altered. One can therefore safely assume that it is always 1 and ignore it. However, this is not true when using perspective projections.
Perspective projection
Another type of transformation, of importance in 3D computer graphics3D computer graphics
3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...
, is the perspective projection. Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection. This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer.
The simplest perspective projection uses the origin as the center of projection, and z = 1 as the image plane. The functional form of this transformation is then
; . We can express this in homogeneous coordinatesHomogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
as:
After carrying out the matrix multiplication, the homogeneous component wc will, in general, not be equal to 1. Therefore, to map back into the real plane we must perform the homogeneous divide or perspective divide by dividing each component by wc:
More complicated perspective projections can be composed by combining this one with rotations, scales, translations, and shears to move the image plane and center of projection wherever they are desired.
See also
- 3D projection3D projection3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.-...
- Transformation (geometry)
- Translation matrix
- Rotation matrix
- Scaling (geometry)Scaling (geometry)In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...
External links
- The Matrix Page Practical examples in POV-RayPOV-RayThe Persistence of Vision Raytracer, or POV-Ray, is a ray tracing program available for a variety of computer platforms. It was originally based on DKBTrace, written by David Kirk Buck and Aaron A. Collins. There are also influences from the earlier Polyray raytracer contributed by its author...
- Reference page - Rotation of axes
- Linear Transformation Calculator
- Transformation Applet - Generate matrices from 2D transformations and vice versa.